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Mirrors > Home > MPE Home > Th. List > metn0 | Structured version Visualization version GIF version |
Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
metn0 | ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 21945 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
2 | frel 5963 | . . . . 5 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → Rel 𝐷) | |
3 | reldm0 5264 | . . . . 5 ⊢ (Rel 𝐷 → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) | |
4 | 1, 2, 3 | 3syl 18 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ dom 𝐷 = ∅)) |
5 | fdm 5964 | . . . . . 6 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → dom 𝐷 = (𝑋 × 𝑋)) | |
6 | 1, 5 | syl 17 | . . . . 5 ⊢ (𝐷 ∈ (Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
7 | 6 | eqeq1d 2612 | . . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → (dom 𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
8 | 4, 7 | bitrd 267 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ (𝑋 × 𝑋) = ∅)) |
9 | xpeq0 5473 | . . . 4 ⊢ ((𝑋 × 𝑋) = ∅ ↔ (𝑋 = ∅ ∨ 𝑋 = ∅)) | |
10 | oridm 535 | . . . 4 ⊢ ((𝑋 = ∅ ∨ 𝑋 = ∅) ↔ 𝑋 = ∅) | |
11 | 9, 10 | bitri 263 | . . 3 ⊢ ((𝑋 × 𝑋) = ∅ ↔ 𝑋 = ∅) |
12 | 8, 11 | syl6bb 275 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 = ∅ ↔ 𝑋 = ∅)) |
13 | 12 | necon3bid 2826 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 × cxp 5036 dom cdm 5038 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 Metcme 19553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-met 19561 |
This theorem is referenced by: (None) |
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